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The asymptotic decay of the magnitude of the Fourier transform of a function appears always to be determined by its continuity properties as follows, with examples given in Fig. 1:

  • Continuous integral, -20 dB / decade;
  • Continuous function, -40 dB / decade;
  • Continuous 1st derivative; -60 dB / decade;
  • ...

How can this be expressed mathematically in a rigorous way, and proved?

enter image description here
Figure 1. dB magnitude of the Fourier transform of a function as function of the base-10 logarithm of frequency for (from slowest to fastest decay) a rectangular pulse (turquoise), a single lobe of a cosine (purple), and Hann function (blue).

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I suppose it would be sufficient to show that a discontinuous function in the time domain decays as $1/\omega$ in the frequency domain (e.g., $\textrm{rect}\Leftrightarrow\textrm{sinc}$), and then use the fact that each integration in the time domain adds a multiplicative factor of $1/j\omega$ in the frequency domain.

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  • $\begingroup$ Yes, or start with a sum of shifted and scaled Dirac deltas and show that its spectrum never settles. Or is periodic if the delays are rational numbers. $\endgroup$ – Olli Niemitalo Aug 7 at 10:01

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